# L2 Regularization is Much More Magical That Most People Think

### An untaught advantage of L2 regularization that most data scientists don't know.

Almost every tutorial/course/blog mentioning L2 regularization I have seen talks about just one thing:

*L2 regularization is a machine learning technique that avoids overfitting by introducing a penalty term into the model’s loss function based on the squares of the model’s parameters.*

*In classification tasks, for instance, increasing the effect of regularization will produce simpler decision boundaries.*

Of course, the above statements are indeed correct, and I am not denying that.

In fact, we can also verify this from the diagram below:

In the image above, as we move to the right, the regularization parameter increases, and the model creates a simpler decision boundary on all 5 datasets.

*Coming back to the topic…*

However, what disappoints me the most is that most resources don’t point out that **L2 regularization is a great remedy for multicollinearity.**

Multicollinearity arises when two (or more) features are highly correlated OR two (or more) features can predict another feature:

When we use L2 regularization in linear regression, the algorithm is also called **Ridge regression**.

But how does L2 regularization eliminate multicollinearity?

Today, let me provide you a demonstrative intuition into this topic, which will also explain **why “ridge regression” is called “ridge regression.”**

Let’s begin!

#### Dummy dataset

For demonstration purposes, consider this dummy dataset of two features:

As shown above, we have intentionally made `featureB`

highly correlated with `featureA`

. This gives us a dummy dataset to work with.

Going ahead, we shall be ignoring any intercept term for simplicity.

### Linear regression without L2 penalty

During regression modeling, the goal is to determine those specific parameters (θ₁, θ₂), which minimizes the residual sum of squares (RSS):

So how about we do the following:

We shall plot the RSS value for many different combinations of (θ₁, θ₂) parameters. This will create a 3D plot:

x-axis → θ₁

y-axis → θ₂

z-axis → RSS value

Next, we shall visually assess this plot to locate those specific parameters (θ₁, θ₂) that minimize the RSS value.

Let’s do this.

Without the L2 penalty, we get the following plot (*it’s the same plot but* *viewed from different angles*):

Did you notice something?

The 3D plot has a valley. There are multiple combinations of parameter values (θ₁, θ₂) for which RSS is minimum.

**Thus, obtaining a single value for the parameters (θ₁, θ₂) that minimize the RSS is impossible.**

### Linear regression with L2 penalty

When using an L2 penalty, the goal is to minimize the following:

Creating the same plot again as we did above, we get the following:

Did you notice something different this time?

As depicted above, using L2 regularization removes the valley we saw earlier and provides a global minima to the RSS error.

**Now, obtaining a single value for the parameters (θ₁, θ₂) that minimizes the RSS is possible.**

Out of nowhere, L2 regularization helped us eliminate multicollinearity.

#### Why the name “ridge regression”?

In fact, this is where “ridge regression” also gets its name from — it eliminates the ridge in the **likelihood function** when the L2 penalty is used.

Of course, in the demonstrations we discussed earlier, we noticed a valley, not a ridge.

However, in that case, we were considering the residual sum of **error** — something which is minimized to obtain the optimal parameters.

Thus, the error function will obviously result in a valley.

If we were to use likelihood instead — something which is maximized, it would (somewhat) invert the graph upside down and result in a ridge instead:

Apparently, while naming the algorithm, the likelihood function was considered.

And that is why it was named “ridge regression.”

Pretty cool, right?

When I first learned about this some years back, I literally had no idea that such deep thought went into naming ridge regression.

Hope you learned something new today :)

If you want to learn about the probabilistic origin of L2 regularization, check out this article: The Probabilistic Origin of Regularization.

👉 Over to you: What are some other advantages of using L2 regularization?

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